The Intimate Relation between Mathematics and Physics

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The Intimate Relation between Mathematics and Physics

Overview

Since the 1960s physics has seen a rebirth of the use of advanced mathematics. Much of this revival occurred after the study of black holes was greatly expanded in the 1960s and 1970s by the English scientists Stephen Hawking (1942- ) and Roger Penrose (1931- ). In the 1980s physicists' 10-dimensional superstring theories received another mathematical boost from tools developed by Edward Witten (1951- ) and others.

Background

Physics and mathematics have always enjoyed a close relationship, beginning in the Renaissance with Johannes Kepler's (1571-1630) 1609 discovery of the three laws of planetary orbits. In 1687 Isaac Newton (1642-1727) introduced the theory of gravity. James Clerk Maxwell (1831-1879) was able to unify the forces of electricity and magnetism in 1865 with the theory of electromagnetism. In the twentieth century mathematical theories from the fields of geometry were instrumental in constructing Albert Einstein's (1879-1955) theory of general relativity as well as in the later development of superstring theory. All of these theories have been predicated upon the prior development of mathematical techniques that had been invented for pure and applied purposes.

In the late seventeenth century Isaac Newton could not have developed the theory of gravity without calculus, a set of mathematical techniques he had developed for studying rates of change. (Calculus was also developed independently by the German mathematician and physicist Gottfried Leibniz (1646-1716.) The definition of gravity underwent another significant revision in 1916 when Albert Einstein showed that gravity could be interpreted as curvatures of space and time. But Einstein could never have developed his theory, now called general relativity, without the non-Euclidean geometry developed by the German mathematician Bernhard Riemann (1826-1866). Riemann's geometry system, developed in 1854, was able to handle descriptions of space where curves predominate and all lines must eventually meet. This was an entirely new way of describing space that the 2000-year-old Euclidean system could not handle.

In the twentieth century several scientists, including Niels Bohr (1885-1962) and Erwin Schrödinger (1887-1961), developed quantum mechanics, which describes the structure of atoms with great precision. Quantum mechanics deals with the microscopic world by treating particles as both particles and waves. Mathematics became increasingly important to physicists during the latter half of the twentieth century, usually because the physical objects under investigation were inaccessible to experimental physics. These objects are as large as black holes and as small as the tiny strings and branes of superstring theory.

Impact

Black Holes

Black holes were initially thought to be strange quirks of Einstein's general relativity theory. These objects are incredibly dense and have a gravity so strong that neither light nor matter can escape. Since they can never be observed directly, descriptions of their shape, size, temperature, and mass remain almost wholly mathematical. (Black hole masses can be calculated relatively easily by observing their effects on surrounding matter.)

In the 1960s Stephen Hawking and Roger Penrose collaborated to study the centers of black holes, regions known as singularities, where time and space become so warped and twisted that they cease to have meaning under normal physical laws. Penrose and Hawking showed that singularities were possible and that under certain conditions would have to be formed. After this discovery, black holes, whose study had formerly been a rather esoteric field, suddenly became a hot subject for many theoretical physicists.

Because black holes seemed to suck in everything and never release anything, they appeared to violate certain physical laws. Then in 1973 Stephen Hawking showed that black holes actually radiate a tiny amount of heat. Hawking proved this by combining mathematics and theories from quantum physics, general relativity, and the laws of thermodynamics—the first time these theories had ever been used simultaneously.

Penrose Tiling and Quasi-Crystals

Penrose also made significant contributions to the field of geometry. In 1974 he showed that nonrepeating patterns could be made out of just two repeating geometric figures. These shapes, called tiles, could be arranged on a flat surface without ever encountering a repeating pattern. In his honor these patterns are now called Penrose tiles. For a decade, Penrose tiles seemed to be an interesting mathematical proof without a real-world application. Then in 1984 a crystalline alloy of aluminum and manganese was discovered that arranged itself into Penrosetiling patterns. These crystals are called quasi-crystals since they do not exhibit the repeating molecular structure of regular crystals.

The Shape of the Universe

The Big Bang theory is currently the best description of the universe's history. It suggests that the universe began as a small, compressed point of matter—possibly a singularity similar to those studied by Penrose and Hawking—that exploded violently outward to form everything we see today. Since we cannot observe the universe in its earliest days—no matter how far we look out with telescopes—describing our universe's birth is a problem for both mathematics and physics. Astrophysicists and cosmologists are also curious about the fate of the universe: Will it continue racing and expanding outward, will the expansion come to a stop, or will it collapse back upon itself? Most observations to date suggest that the universe will keep expanding; whether the expansion will slow to zero or continue forever, however, is still unknown.

A theory first proposed in 1980 by Alan Guth of the Massachusetts Institute of Technology, called inflation theory, predicts another outcome. Inflation theory states that the universe underwent an exponential expansion when it was about the size of a pea. This expansion will eventually come to a stop, making the universe "flat" and obeying Euclidean geometry. Most calculations of the universe's total mass do not support this view, however. Instead, astronomers' observations show that the universe does not have enough mass to halt the runaway expansion and will, therefore, continue expanding forever.

Inflation theory might work, however, if scientists could account for this apparent lack of matter. One theory proposes the existence of "dark matter," which is matter that does not glow with enough energy to be seen from earth. A large-enough amount of dark matter could halt the expansion. The other potential solution involves a mysterious form of energy often called the cosmological constant. It was a "fudge factor" introduced by Einstein to make the theory of general relativity account for observations regarding the unchanging size of the universe. He later retracted the cosmological constant, calling it the biggest mistake of his life, when Edwin Hubble (1889-1953) showed that the universe was indeed expanding. Today cosmologists have resurrected the cosmological constant in a different form, believing that it could be an undiscovered form of energy that will slow the universe's expansion, keeping it flat and in agreement with inflationary theory.

Cosmologists who accept that the universe does not contain enough mass or energy to halt expansion say that our universe is "negatively curved." Mathematicians have shown that many negatively curved spaces with hyperbolic geometry can fold up in ways that could still contain a finite universe. These shapes also give rise to some rather interesting conjectures, one of which is that you could travel in a straight line across the universe and eventually end up at your starting point. Another is that we could conceivably look out and see our own Milky Way galaxy at a young age after its light had traveled around the entire universe. Proving this theory would require very detailed observations of the skies. In 2000 NASA's Microwave Anisotropy Experiment satellite will begin to make some of these observations.

String Theory

Superstring theory had its beginnings in the 1970s. At that time, some physicists were dissatisfied with contemporary theories that treated subatomic particles as points. They encountered too many complexities when using a dimensionless point to describe fundamental particles (these were the same kinds of difficulties encountered by Penrose and Hawking in their study of singularities.) These physicists tried instead to come up with a method for treating the particles as loops, or strings. The vibrational patterns of these loops, in turn, determined their characteristics and how they formed larger particles, such as quarks, electrons, and protons. But the theory, simply called string theory back then, was not very successful.

In the 1980s the theory was reborn as supersymmetric 10-dimensional string theory, or superstring theory for short. Theoretical physicists found that using one-dimensional strings in a 10-dimensional universe showed promising results. The theory blossomed with investigations by Edward Witten and others. This period in the mid-1980s is now known as the first golden age of string theory, during which physicists were able to make giant leaps forward. Since then the development of string theory has slowed its rapid pace of advancement.

Unfortunately, superstring theory deals with particles so small that physicists have little hope of ever detecting them or even building a large-enough detector. A particle accelerator large enough for the task would need to be larger than the solar system. This difficulty leaves many scientists leery of superstring theory. If the theory remains unverifiable by experiment, how can it ever be proved? In addition, a theory that states that there are 10 dimensions, six of which are curled up so tightly that we cannot see them, rubs against today's four-dimensional viewpoint, which involves three dimensions in space and one of time. String theory supporters remain convinced of their theory's veracity because it fits so well with everything else we know. They believe that superstring theory is one of the few instances in which theory has been able to leap ahead of experiment. Also, since the mathematics involved are so advanced, many string theorists admit that they still do not understand all the implications of the mathematics they have discovered. As Witten has said, it's twenty-first-century mathematics that miraculously landed in the twentieth.

The Seiberg-Witten Equations

Transfers between mathematics and physics have not just been in one direction. At the end of 1994 physics made a huge contribution to the mathematical field of topology when Ed Witten and Nathan Seiberg found that some of the mathematics they had developed to study singularities was applicable to the study of manifolds, which are descriptions of how space and curves can be deformed and stretched without changing certain properties. The equations caused a revolution in the study of manifolds and gave mathematicians an entirely new classification scheme. This development, in turn, has been important for studying the effects of gravity on space and time.

PHILIP DOWNEY